SIAM J. A MATH c Vol. 70, No. 6, pp. 1922–1952djelatnici.unizd.hr/~makosor/SIAM_stent.pdf · their design and performance. Even though a lot of attention has been devoted in cardiovascular

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SIAM J. APPL. MATH. c© 2010 Society for Industrial and Applied MathematicsVol. 70, No. 6, pp. 1922–1952

MATHEMATICAL MODELING OF VASCULAR STENTS∗

J. TAMBACA† , M. KOSOR‡ , S. CANIC‡ , AND D. PANIAGUA, M.D.§

Abstract. We present a mathematical model for a study of the mechanical properties of en-dovascular stents in their expanded state. The model is based on the theory of slender curved rods.Stent struts are modeled as linearly elastic curved rods that satisfy the kinematic and dynamic con-tact conditions at the “vertices” where the struts meet. This defines a stent as a mesh of curved rods.A weak formulation for the stent problem is defined and a finite element method for a numericalcomputation of its solution was developed. Numerical simulations showing the pressure-displacement(axial and radial) relationship for the entire stent are presented. From the numerical data and fromthe energy of the problem we deduced an “effective” pressure-displacement relationship of the law ofLaplace-type for the mechanical behavior of stents, where the proportionality constant in the Laplacelaw was expressed explicitly in terms of the geometric and mechanic properties of a stent.

Key words. stents, mathematical modeling, theory of elasticity, curved rod model

AMS subject classifications. 74K10, 74K30, 74B05, 74S05

DOI. 10.1137/080722618

1. Introduction. A stent is a mesh tube that is inserted into a natural con-duit of the body to prevent or counteract a disease-induced localized flow constric-tion. Endoluminal stents are used in the cardiovascular system (coronary arteries,pulmonary arteries, aorta, large systemic veins and arteries, etc.) as well as in thetracheobronchial, biliary, and urogenital systems. They play a crucial role in thetreatment of coronary artery diseases (CAD). Coronary artery disease (or clogging,or stenosis of a coronary artery) is the major cause of heart attack, the leading causeof death in the United States. One person dies about every minute from a coronaryevent. Treatment of CAD entails inserting a catheter with a mounted balloon whichis inflated to widen the lumen of a diseased artery (the area occupied by blood) andrestore normal blood flow. This procedure is called balloon angioplasty. To propthe arteries open, a stent is inserted at the location of the narrowing. See Figure 1.Clinical and computational studies show that performance of coronary stents depends,among other things, on the geometrical properties of a stent, such as the number ofstent struts, the strut width and thickness, and the geometry of the cross section ofeach stent strut, [21, 14, 6, 2, 12, 15, 11]. At the same time these geometric propertiesdetermine the overall mechanical properties of a stent.

By now, there is a large number of stents with different geometrical and mechan-ical features available on the market. Knowing the mechanical properties of a stent is

∗Received by the editors April 28, 2008; accepted for publication (in revised form) November 16,2009; published electronically February 19, 2010.

http://www.siam.org/journals/siap/70-6/72261.html†Department of Mathematics, University of Zagreb, Bijenicka 30, 10000 Zagreb, Croatia

([email protected]). Research supported in part by MZOS-Croatia under grant 037-0693014-2765and by the NSF/NIH under grant DMS-0443826.

‡Department of Mathematics, University of Houston, Houston, 4800 Calhoun Road, Houston,TX 77204-3476 ([email protected], [email protected]). The work of the second author wassupported through NSF/NIH under grant DMS-0443826. The work of the third author was partiallysupported by the NSF under grant DMS-0337355, Texas Higher Education Board ARP 003652-0051-2006, NSF/NIH under grant DMS-0443826, and UH GEAR-2007 grant.

§Co-Director, Cardiac Catheterization Laboratories Michael E. DeBakey Veterans Medical Center,Assistant Professor of Medicine Baylor College of Medicine, St. Luke’s Episcopal Hospital/TexasHeart Institute, Houston, TX 77030-2604.

1922


MATHEMATICAL MODELING OF VASCULAR STENTS 1923

Fig. 1. Deployment of a coronary stent.

important in determining what pressure loads a stent can sustain when inserted in anative artery. Different medical applications require stents with different mechanicalproperties. As noted in [8], the therapeutic efficacy of stents depends exclusively ontheir mechanical properties.

Numerical studies of mechanical properties of vascular stents are a way to improvetheir design and performance. Even though a lot of attention has been devoted incardiovascular literature to the use of endovascular prostheses over the past 10–15years, the engineering and mathematical literature on the numerical studies of themechanical properties of stents is not nearly as rich. Various issues in stent designand performance are important depending on the questions asked. They range fromthe study of large deformations that a stent undergoes during balloon expansion, forwhich nonlinear elasticity and plasticity need to be considered, all the way to thesmall deformations exhibited by an already expanded stent inserted in an artery, forwhich linear elasticity might be adequate. A range of issues has been studied in [5, 13]and the references therein, involving several different approaches. Most approaches,however, use commercial software packages based on the three-dimensional (3D) finiteelement method (FEM) structure approximations which may be computationally veryexpensive.

In this manuscript we present a novel mathematical model of a stent defined as amesh of one-dimensional curved rods. A curved rod model is a one-dimensional (1D)approximation of a three-dimensional rod-like structure [9, 10]. In contrast with thebeam theory which only takes into account transverse deformations of a beam in thedirection of the force, the curved rod theory which was used in this manuscript tomodel the mechanical properties of stent struts takes into account deformations inall three spatial directions. This is particularly important in stent strut modeling inwhich, as shown in this manuscript, deformations in all three spatial directions takeplace and may be of the same order of magnitude, depending on the type of forcing.By prescribing the kinematic and dynamic contact conditions at the points where thecurved struts meet (vertices), a stent is defined as a mesh of curved rods with theappropriate geometric properties.

Following a weak formulation of the stent problem as a mesh of 1D curved rods,we present the details of the development of a FEM approximation of the modelequations in section 5. Our approach, based on the curved rod theory, simplifies thecomputation of the mechanical properties of stents depending on their geometric andmechanic parameters (in the realm of small deformations), thereby enabling a largenumber of simulations corresponding to different combinations of the geometric andmechanic parameters. (A comparison between a 3D and the 1D curved rod modelperformance is presented in Appendix B.)

We show numerical results indicating the overall, effective mechanical propertiesof stents depending on their geometric structure and on the struts’ mechanical prop-erties in section 6. We focus on a series of scenarios corresponding to an already


1924 J. TAMBACA, M. KOSOR, S. CANIC, AND D. PANIAGUA, M.D.

expanded stent undergoing small deformations due to a pressure load exerted to theinterior and/or exterior stent surface. In particular, in section 7 we studied a rela-tionship between the applied pressure and the displacement (radial and axial) as afunction of the following parameters:

• Geometric parameters: t—thickness of stent struts, w—width of stent struts,nO—number of vertices in the circumferential direction, nL + 1—number ofvertices in the axial direction, R—stent reference radius (expanded), L—stentreference length (expanded).

• Mechanic parameters: E—struts Young’s modulus, μ—struts shear modulus.This enabled us to generate a set of data that we used to deduce the overall, effec-tive pressure-displacement relationship of the law of Laplace-type for the mechanicalbehavior of stents, where the proportionality constant in the Laplace law depends ex-plicitly on the geometric and mechanic properties of a stent. Then, from the formula-tion of the stent problem as a minimization of an energy functional, obtained from itsweak formulation, and by using simple geometric arguments, we were able to recoveranalytically the numerically derived effective pressure-displacement relationships.

The simple and elegant effective pressure-displacement formulas can be used, forexample, to mimic stent compliancy in the more complex hemodynamics studies offluid-structure interaction between blood flow and a stented artery, and, for example,to estimate the pressure load that a stent can sustain in the realm of small deformation(e.g., less than 10%), which is useful for manufacturing or choosing the right stent fora given application.

2. Stent frame geometry. During catheter deployment, high-precision lasercut stents are expanded using a balloon that has been premounted on a catheter. SeeFigure 1. In an expanded configuration the stent assumes a certain radius R andlength L which will be called the reference radius and reference length, respectively.We are interested in describing the “stress-strain” relationship between the pressureload exerted on the outer and inner surfaces of a stent and the change in the diameterand length of a stent, for a given set of mechanical and geometrical stent parameters.Thus, we focus on the following problem: Given a set of parameters determining thegeometric and mechanic properties of a stent, find the pressure-displacement relation-ship for the overall stent structure.

We consider a stent to be a three-dimensional elastic body defined as a unionof three-dimensional struts (see Figures 2 and 3). The struts, which are three-dimensional, will be modeled by a curved rod model. A curved rod model is aone-dimensional approximation of a “thin” three-dimensional curved elastic struc-ture given in terms of the arc-length of the middle curve of the rod as an unknownvariable. The cross section of a rod representing each stent strut is assumed to berectangular, of width w and thickness t. Thickness t is the dimension of a strut in thenormal direction to the strut, as shown in Figure 4, which corresponds to the radialdirection of the global cylindrical stent geometry. Struts themselves are assumed tobe linearly elastic, with the elastic parameters given by the Lame constants λ and μ,or, equivalently, by the Young’s modulus of elasticity E and the shear modulus μ.

Stent struts form a frame of diamonds with nO vertices in the circumferentialdirection and nL+1 vertices in the longitudinal direction (see Figure 3). The presen-tation in this manuscript will focus on the uniform stent geometry, where all the strutsare of equal length. This is, however, not required for the development of the theo-retical and numerical methods described below, as they can be generalized to stentsof arbitrary geometry with struts of different lengths. We have, in fact, used this fea-



Fig. 2. A photograph of a stent with uniform geometry.

Fig. 3. A stent with nO = 6 and nL = 5.

ti, jk

ni, jk

bi, jk

Pi, jk

w

t

Fig. 4. A strut of width w and thickness t. The picture also shows the image of the middlecurve parameterization P k

i,j and the local basis consisting of the tangent vector tki,j , the normalvector nk

i,j, and the bi-normal vector bki,j , defined in (2.3). The normal vector nki,j corresponds to

the radial direction in a stent.

Fig. 5. The figure shows the angle formed by a vertex of a stent, the center of the circular crosssection, and an adjacent vertex on the stent, denoted by φ = 2π/nO.



vi, j

vi, j 1

vi 1, j 1 vi, j 1

vi 1, j 1

Pi 1, j 11

Pi, j 10 Pi, j

1

Pi, j0

Fig. 6. The figure shows the nomenclature for the incoming and outgoing struts of the vertex vi,j.

vi, j

vi, j 1 Pi, j0

Si, j0

Fig. 7. Curved stent strut.

ture of our numerical method to test the mechanical properties of a nonuniform stentdesigned for a transcatheter implantation of an aortic valve prosthesis [16, 17, 20].The presentation of the methods, however, is much clearer if we assume throughoutthis paper that our stent has a uniform geometry.

To describe the basic equations modeling the mechanical properties of stents, weintroduce the following notation. Suppose that a transverse cross section of a stentis taken. Then the angle formed by a vertex of a stent, the center of the circularcross section, and an adjacent vertex on the same circumference of the stent, will bedenoted by φ = 2π/nO. See Figure 5. The vertices on the adjacent circular crosssection are shifted by the angle φ/2. With this notation, the vertices on the stent canbe described by

vi,j =

(R cos((i − 1)φ+ (j − 1)φ/2), R sin((i − 1)φ+ (j − 1)φ/2), (j − 1)

L

nL

)T

,

where j = 1, . . . , nL + 1 denotes the indices of the vertices in the longitudinal stentdirection, and i = 1, . . . , nO denotes the indices of the vertices in the circumferentialstent direction. Two incoming and two outgoing struts exist for all interior vertices.The incoming struts of a vertex vi,j are those connecting the vertices shifted by theangle ±φ/2 at the level j − 1 with the vertex vi,j . Similarly, the outgoing struts arethose connecting the vertex vi,j by the two vertices shifted by the angle ±φ/2 at thelevel j + 1. See Figure 6.

Struts of a high-precision laser cut stainless steel stent are not straight, but curvedand located on the cylinder of radius R. To write the equations for the curved stentstruts we take a cord connecting the two vertices that define a strut, and then projectthe cord to the cylinder of radius R. See Figure 7. More precisely, denote by Rk

i,j ,k = 0, 1, the two outgoing struts emerging from the vertex vi.j , and connecting to thevertices shifted by ±φ/2 at the level j+1. Then the cords (straight lines) connectingvi.j to the vertices shifted by ±φ/2 at the level j + 1 can be parameterized as

Ski,j(s) = svi,j + (1− s)v((i−1−k) modnO)+1,j+1, s ∈ [0, 1],

i = 1, . . . , nO, j = 1, . . . , nL, k = 0, 1,(2.1)

where k = 0, 1 corresponds to the struts Rki,j , k = 0, 1. The middle curve of the



curved stent struts Rki,j can be expressed via the parameterization

P ki,j : [0, 1] → R

3

(see Figure 4), where

P ki,j(s) = NSk

i,j(s), s ∈ [0, 1], i = 1, . . . , nO, j = 1, . . . , nL, k = 0, 1.(2.2)

Here N is the operator that lifts the cord up to the cylinder of radius R:

Nv = Pv +Rv − Pv

‖v − Pv‖ ,

where P denotes the orthogonal projector on e3 in R3 with the standard scalar prod-

uct, and {e1, e2, e3} is the standard orthonormal basis of R3.The distance of the endpoints of a strut in R

3 is given by

le =√(R−R cos (φ/2))2 + (R sin (φ/2))2 + (L/nL)2.

Note that this number is different from the strut length ls but it is a good approxi-mation of the strut length for the slightly curved struts.

We are now in a position to introduce a parameterization of the three-dimensionalstent struts Rk

i,j that define a three-dimensional stent Ω. Introduce a local basis at

each point on the middle curve of strut Rki,j (see Figure 4):

tki,j(s) =(P k

i,j)′(s)

‖(P ki,j)

′(s)‖ , nki,j(s) =

(I − P )P ki,j(s)

‖(I − P )P ki,j(s)‖

, bki,j(s) = tki,j(s)×nki,j(s)(2.3)

for s ∈ [0, 1]. Then the three-dimensional strut Rki,j can be parameterized by

Φki,j(s1, s2, s3) = P k

i,j(s1) + s2nki,j(s) + s3b

ki,j(s),

where P ki,j is defined by (2.2). The parameterizationΦk

i,j maps the set [0, 1]×[−t/2, t/2]× [−w/2, w/2] into R

3. Notice that the normal vector points in the radial direction.A stent Ω can now be defined as a three-dimensional domain which is a union of

stent struts Rki,j parameterized by Φk

i,j :

Ω = ∪nO

i=1 ∪nL

j=1 ∪1k=0Φ

ki,j ([0, 1]× [−t/2, t/2]× [−w/2, w/2]) .(2.4)

The interior stent surface of a stent is defined by

ΓI = ∪nO

i=1 ∪nL

j=1 ∪1k=0Φ

ki,,j ([0, 1]× {−t/2} × [−w/2, w/2]) ,

and the exterior stent surface by

ΓE = ∪nO

i=1 ∪nL

j=1 ∪1k=0Φ

ki,j ([0, 1]× {t/2} × [−w/2, w/2]) .

In this section we have described a geometry of the stent frame. This will beused in the following sections where we focus on modeling the mechanical propertiesof stents. In particular, we are interested in the mechanical response of an expandedstent Ω under the pressure load exerted on either the exterior surface ΓE or theinterior surface ΓI . As we shall see later in this manuscript, we will be assuming small



deformations and small deformation gradients so that the theory of linear elasticitycan be used to model a stent as a homogeneous, isotropic linearly elastic body byassuming that each stent strut satisfies the same mechanical properties. We will beinterested in a pure traction problem. Namely, if a pressure load is applied to, e.g.,the exterior surface of Ω, the remaining boundary will be assumed to be force free. Anecessary condition for the existence of a solution to a pure traction problem is givenby the requirement that the total force and the total moment applied to the structurebe equal to zero (see, e.g., [4]). Moreover, the problem admits nonunique solutions: asolution is unique up to the infinitesimal rigid displacement a+ b× x, where x ∈ R

3,and a, b ∈ R

3 are arbitrary. Thus, in order to have a well-posed problem, we look for afunction u that is orthogonal in the L2-sense, to all infinitesimal rigid displacements,namely, such that ∫

Ω

u(x) · (a + b× x)dx = 0 ∀a, b ∈ R3.

These are the main ideas that will be followed in the subsequent sections. Addition-ally, to simplify analysis and numerical simulation of the entire stent structure, weapproximate each three-dimensional “thin” stent strut by a one-dimensional curvedrod model and use this approach to study the overall mechanical properties of astent as a collection of one-dimensional curved rods. This approach is new in thestudy of mechanical properties of endovascular stents. It simplifies greatly numericalsimulation of mechanical properties of stents, allowing complicated structures to besimulated in a very short time frame.

In the next section we describe the curved rod model for an approximation of asingle stent strut and later define the appropriate boundary conditions at the pointswhere the struts meet (vertices) to derive a mathematical model for the entire stentas a frame structure consisting of curved rods.

3. The curved rod model for a single stent strut. The curved rod model isa one-dimensional model that approximates a three-dimensional rod-like structure tothe ε2 accuracy, where ε is the ratio between the largest dimension of the cross sectionand the length of a rod. For a derivation and mathematical justification of the curvedrod model see [9] and [10]. A numerical comparison between the perfomance of theone-dimensional curved rod model and a three-dimensional linearly elastic rod-likestructure is presented in Appendix B. In general, the behavior of a three-dimensionalrod-like elastic body is approximated by the behavior of its middle curve and of itscross sections. In the curved rod model, the cross sections behave approximately asinfinitesimal rigid bodies that remain perpendicular to the deformed middle curve.

More precisely, let P : [0, �] → R3 be the natural parameterization of the middle

curve of the rod of length � (‖P ′(s)‖ = 1, s ∈ [0, �]). Then the curved rod modelcan be formulated as a first-order system of differential equations for the followingunknown functions:

• u : [0, �] → R3, the displacement of the middle curve of the rod;

• ω : [0, �] → R3, the infinitesimal rotation of the cross section of the rod;

• q : [0, �] → R3, the contact moment; and

• p : [0, �] → R3, the contact force.

(Here � corresponds to the strut length, denoted by ls.) For a given line force densityf , the equations of the curved rod model can be written as (see [18])

p′ + f = 0,(3.1)

q′ + t× p = 0,(3.2)



describing the balance of contact force and contact moment, respectively, with

ω′ −QH−1QTq = 0,(3.3)

u′ + t× ω = 0,(3.4)

describing the constitutive relations for a curved, linearly elastic rod. Here t is theunit tangent to the middle curve, Q = (t,n, b) is the orthogonal matrix containingthe tangent vector t and vectors n and b that span the normal plane to the middlecurve (Q describes the local basis at each point of the middle curve), and

H =

⎡⎣ μK 0 0

0 EIb 00 0 EIn

⎤⎦ ,

where E = μ 3λ+2μλ+μ is the Young’s modulus of the material, In and Ib are moments of

inertia of a cross section and μK is the torsion rigidity of the cross section. Therefore,H describes the elastic properties of the rod and the geometry of the cross section.This model can also be obtained by a linearization of the Antman–Cosserat rod model(see [1]) under the material restriction of inextensibility and unshearability of a rod.

Equation (3.4) is a condition that requires that the middle line is approximatelyinextensible and that allowable deformations of the cross section are approximatelyorthogonal to the middle line. This condition has to be included in the solution spacefor the weak formulation of problem (3.1)–(3.4) (pure traction problem for a singlecurved rod). Thus, introduce the space

V ={(v, w) ∈ H1(0, �)6 : v′ + t× w = 0

}.(3.5)

Function (u, ω) ∈ V is called a weak solution of problem (3.1)–(3.4) if

∫ �

0

QHQT ω′ · w′ds

=

∫ �

0

f · vds+ q(�) · w(�)− q(0) · w(0) + p(�) · v(�)− p(0) · v(0)(3.6)

holds for all (v, w) ∈ V (notice the difference in the notation between ω and w).Equation (3.6) will be used in the next section to define a weak formulation for theentire stent defined as a collection of curved rods. Boundary conditions appearing in(3.6) will be specified through the contact conditions at the stent vertices.

4. Stent as a collection of curved rods. We recall that in section 2 a stentΩ was defined as a three-dimensional domain which is a union of three-dimensionalstent struts Rk

i.j parameterized by Φki,j ; see (2.4). To model the mechanical behavior

of a stent as a collection of one-dimensional linearly elastic, homogeneous, isotropiccurved rods, we parameterize the struts using the one-dimensional parameterizationsP ki,j of the struts’ middle curves; see (2.2). Now a stent can be defined as a union of

one-dimensional parameterizations as follows:

F =

nO⋃i=1

nL⋃j=1

1⋃k=0

P ki,j([0, 1]).

Note that parameterizations P ki,j are not arc-length parameterizations which is nec-

essary for the formulation of the curved rod model (3.1)–(3.4). Nevertheless, they



uniquely determine the middle curves of the stent struts and imply the existence ofthe arc-length parameterizations. Finding the arc-length parameterization in this caseis a difficult task which, as we shall see later in section 4.1, is not necessary for thefinal formulation of the problem and the numerical method development.

Each of the curved rods approximating the stent struts Rki,j satisfies a set of

equations of the form (3.1)–(3.4); namely, for each Rki,j we have

(pki,j)

′ + fk

i,j = 0,

(qki,j)

′ + tki,j × pki,j = 0,(4.1)

(ωki,j)

′ −Qki,jH

−1(Qki,j)

T qki,j = 0,

(uki,j)

′ + tki,j × ωki,j = 0,

on 〈0, ls〉; here tkij and Qki,j denote the unit tangent vector function and the rotation

matrix function for the curved rod Rki,j .

At the vertices where the curved rods meet, the kinematic and dynamic contactconditions determine the boundary condition for each curved rod in the stent framestructure. The kinematic contact condition describes the continuity of the kinematicquantities uk

i,j and ωki,j , stating that the displacement and the infinitesimal rotation

for two struts meeting at a vertex are the same. The dynamic contact condition isthe equilibrium condition requiring that the sum of all contact forces at a vertex andthe sum of all contact moments at a vertex be equal to zero. Thus, for each vertexvi,j the kinematic contact conditions are then given by

u0(i−1)modnO+1,j−1(ls) = u1

imodnO+1,j−1(ls) = u0i,j(0) = u1

i,j(0),(4.2)

ω0(i−1)modnO+1,j−1(ls) = ω1

imodnO+1,j−1(ls) = ω0i,j(0) = ω1

i,j(0),(4.3)

and the dynamic contact conditions are given by

q0(i−1)modnO+1,j−1(ls) + q1

imodnO+1,j−1(ls) = q0i,j(0) + q1

i,j(0),(4.4)

p0(i−1)modnO+1,j−1(ls) + p1

imodnO+1,j−1(ls) = p0i,j(0) + p1

i,j(0),(4.5)

for i = 1, . . . , nO, j = 1, . . . , nL + 1 with the convention that the quantity is removedfor nonexistent indices corresponding to the end vertices vi,1 and vi,nL+1.

To define a weak formulation for the stent frame problem, introduce the followingfunction space:

VF ={(v0

1,1, w01,1, . . . , v

1nO,nL

, w1nO,nL

) : (vki,j , w

ki,j) ∈ V k

i,j & (4.2), (4.3) hold},

where V ki,j are the function spaces (3.5) corresponding to the struts Rk

i,j .Now the weak formulation for the stent frame structure consisting of curved rods

is given by the following.Definition 4.1. Function (u0

1,1, ω01,1, . . . , u

1nO,nL

, ω1nO,nL

) ∈ VF is a weak solu-tion to the stent frame problem (4.1)–(4.5) if

nO∑i=1

nL∑j=1

∑k=0,1

∫ ls

0

Qki,jH(Qk

i,j)T (ωk

i,j)′ · (wk

i,j)′ds =

nO∑i=1

nL∑j=1

∑k=0,1

∫ ls

0

fk

i,j · vki,jds(4.6)

holds for all (v01,1, w

01,1, . . . , v

1nO,nL

, w1nO,nL

) ∈ VF .



Notice again the difference in the notation for the infinitesimal rotation test func-tions wk

i,j and the notation for the infinitesimal rotation solution functions ωki,j . Also

notice that all the intermediate boundary terms on the right-hand side of equation(3.6) cancel out in the formulation (4.6) due to the kinematic and dynamics contactconditions.

The solution to problem (4.6) is not unique. Namely, since only the derivative ofω appears in the weak formulation, the solution will be determined up to a constantω0. Thus, if P is a point on the frame structure, then ω(P ) = ω0 is in the kernel ofthe problem. Furthermore, from the condition u′ + t × ω = 0, with ω constant, onecan solve the equation for u to obtain u(s) = u0 −P × ω0 = u0 + ω0 ×P . Thus, theinfinitesimal rotation of the cross section and displacement of P are unique up to theterm [

ω(P )u(P )

]=

[ω0

u0 + ω0 × P

],

for arbitrary vectors u0, ω0 ∈ R3. This means that the solution is unique up to the

translation and infinitesimal rotation of the frame structure. Thus, as hinted earlier,we will be interested in the solution of (4.6) that satisfies an additional condition∫

F

u(P ) · (a+ b× P )dP = 0 ∀a, b ∈ R3.(4.7)

The frame structure presented in this section is still extremely complex. Themain obstacle for the numerical treatment of the problem of the form (4.6) is theimplementation of the condition in the function spaces V k

i,j that should be satisfied bythe test functions. For this reason, we make a further simplification that incorporatesapproximation of each curved rod by the piecewise straight rods. This approximationhas been mathematically justified in [18] and [19].

4.1. Approximation of a curved rod by piecewise linear rods. Based onthe results in [18] and [19], if we perturb the middle curve of a curved rod by δ in theW 1,∞-norm, the difference between the solution of the original problem W = (ω, u)and the solution of the perturbed problem W δ = (ωδ, uδ) will satisfy the followingestimate:

|W (P )−W δ(P )|L∞ ≤ Cδ.

Thus, it is reasonable to introduce piecewise linear rods that approximate eachcurved rod in the following way. Consider the cords (straight line segments) Sk

i,j ,defined in (2.1), connecting the two vertices defining a curved strut. Divide the cordSki,j into nS equidistant segments, and “lift them up” to the cylindrical surface of

radius R. These points define the new vertices

vk,li,j = N

(l

nSvi,j +

(1− l

nS

)v((i−1−k) modnO)+1,j+1

)= NSk

i,j(l/nS),

i = 1, . . . , nO, j = 1, . . . , nL, k = 0, 1, l = 1, . . . , nS − 1.

Connect the new vertices by straight line segments to define the new straight “struts”approximating the curved strut. Each new linear piece is of length �k,li,j = ‖vk,l

i,j−vk,l−1i,j ‖

and is parameterized by

P k,li,j (s) = vk,l−1

i,j + svk,li,j − vk,l−1

i,j

‖vk,li,j − vk,l−1

i,j ‖ , s ∈[0, �k,li,j

], i = 1, . . . , nO,

j = 1, . . . , nL, k = 0, 1, l = 1, . . . , nS,



where the convention vk,0i,j = vk

i,j and vk,nS

i,j = v((i−1−k) modnO)+1,j+1 is used. Theseparameterizations are natural; i.e., they are given in terms of the arc-length, as re-quired by the curved rod model.

Now we use the fact that the formulation of the curved rod model (3.4) is valid forthe piecewise smooth curves; see [18]. Each linear part can be treated as a straightrod with contact conditions of the same form as (4.2), (4.3) and (4.4), (4.5) at itsends, where there are only two struts intersecting at the new vertices. The weakformulation of this problem is of the same structure as that of problem (4.6), butwith 2nOnL(nS − 1) extra vertices.

Moreover, since all the rods are straight, the equations of equilibrium simplify. Inparticular, if we employ the following notation:

• ut, un, and ub are the tangential, normal, and binormal components of thelocal displacement, and

• τ is the tangential rotation of the cross section (torsion) [18],then the last condition in (3.4) implies that u and ω can be expressed as

u = Q

⎡⎣ utunub

⎤⎦ , ω = Q

⎡⎣ τ

−u′bu′n

⎤⎦ .

Moreover, for straight rods, condition (3.4) implies that ut is constant, i.e., ut ∈ R.Thus, we no longer need to incorporate condition (3.4) in the function spaces V k

i,j

which simplifies greatly the numerical implementation of the model equations.The weak formulation for the problem in which each curved rod is approximated

by iecewise straight rods now reads as follows.Find ((uk,li,j )t, (u

k,li,j )n, (u

k,li,j )b, τ

k,li,j ) ∈ R×H2(0, �k,li,j )×H2(0, �k,li,j )×H1(0, �k,li,j ) such

that

∑i,j,k,l

∫ �k,li,j

0

EIn(uk,li,j )

′′n(v

k,li,j )

′′n + EIb(u

k,li,j )

′′b (v

k,li,j )

′′b + μK(τk,li,j )

′(ψk,li,j )

′ds

=∑i,j,k,l

∫ �k,li,j

0

(fk,li,j )t(v

k,li,j )t + (fk,l

i,j )n(vk,li,j )n + (fk,l

i,j )b(vk,li,j )bds(4.8)

holds for all ((vk,li,j )t, (vk,li,j )n, (v

k,li,j )b, ψ

k,li,j ) ∈ R×H2(0, �)×H2(0, �)×H1(0, �) satisfying

the kinematic contact conditions at all the vertices, namely,• at the original vertices determined by the curved rods:

v0,1i,j (0) = v1,1

i,j (0) = v1,nS

imodnO+1,j−1

(�1,nS

imodnO+1,j−1

)

= v0,nS

(i−1)modnO+1,j−1

(�0,nS


),(4.9)

w0,1i,j (0) = w1,1

i,j (0) = w1,nS

imodnO+1,j−1

(�1,nS

imodnO+1,j−1

)

= w0,nS


(�0,nS


),(4.10)

for each i = 1, . . . , nO, j = 1, . . . , nL; and• at the interior vertices determined by the piecewise linear rods:

v0,li,j(�

0,li,j) = v0,l+1

i,j (0), and w0,li,j(�

0,li,j) = w0,l+1

i,j (0),(4.11)

for each i = 1, . . . , nO, j = 1, . . . , nL, and l = 1, . . . , nS − 1.



Here

fk,l

i,j = Q

⎡⎢⎣

(fk,li,j )t

(fk,li,j )n

(fk,li,j )b

⎤⎥⎦ ,

and E is the Young’s modulus, In and Ib are moments of inertia of a cross section,and μK is the torsion rigidity of a cross section, as defined in section 3.

5. The numerical method. To solve problem (4.8)–(4.11) we designed a nu-merical solver based on the finite element method. We solve problem (4.8)–(4.11) interms of the displacement and rotation at the vertices. Denote them generically byU ∈ R

3 and W ∈ R3. Recall that for each strut the displacement function u and

rotation w are expressed by ut, un, ub, and τ . We approximate these functions by thefollowing polynomials:

ut(s) = Et, un(s) = Bns3 + Cns

2 +Dns+ En,

ub(s) = Bbs3 + Cbs

2 +Dbs+ Eb, τ(s) = 2Cts+Dt.

These polynomials satisfy u′t = 0, i.e., ut ∈ R, and (un, ub, ψ) ∈ H2(0, �)×H2(0, �)×H1(0, �), for an appropriate segment length �.

Let U0, W 0, U �, W � denote the displacement and rotation of the cross section atthe endpoints of each given linear piece. Then, for each U0, W 0, U �, W � we are ableto express vt, vn, vb, and ψ uniquely under the condition that

U0 · t = ut(0) = ut(�) = U � · t.(5.1)

This condition is a consequence of the fact that for straight rods (ut)′ = 0. This

condition should be satisfied by both the unknown functions (solution) and the testfunctions for all linear pieces. To enforce this condition, we employ the approachbased on the Lagrange multipliers. The final linear system is of the form[

A CT

C 0

] [XY

]=

[F0

],(5.2)

where X is a vector of unknown functions X = (U1, W 1, . . . , Unv , W nv )T , with

nv = nO(nL + 1) + 2nOnL(nS − 1) denoting the number of all vertices and Y thevector of Lagrange multipliers. The dimension ofX is equal to six times the number ofall vertices nv, and the dimension of Y is equal to the total number of all linear struts2nOnLnS . The matrix A is the stiffness matrix of the system which is symmetric. Cis the matrix of conditions (5.1).

The solution of problem (4.8)–(4.11) is not unique. The matrix of the system(5.2) is singular with the kernel of dimension 6. As mentioned earlier, to obtain aunique solution, we seek a solution which satisfies the additional condition (4.7). Adiscrete form of this condition can be written as∑

i,j,k

Uk,l

i,j =∑i,j,k

Wk,l

i,j = 0.

To enforce condition (4.7) we use the penalization method.A code written in C++ was developed to implement this approach. We have been

working with frames consisting of 100–250 vertices, so the matrices of the systems areof dimensions up to 2000. The time to solve the problem numerically varies from 0.3to 5 seconds on a server with one Intel Xeon 3.00 GHz processor and 2GB of RAM.



6. Numerical results. In most of the examples below, various configurations ofstents will be exposed to the exterior pressure loads of 0.5 atmospheres. This pressureload is physiologically reasonable since stents are typically oversized by 10% of thenative vessel radius to provide reasonable fixation. If we assume an approximateYoung’s modulus of a coronary artery to be between 105Pa and 106Pa (see [3]), thenusing the law of Laplace one can estimate an approximate pressure exerted by thearterial wall to a coronary stent to be around 0.5 atm.

A series of examples below will investigate how stents respond to uniform versusnonuniform pressure loads, and how their overall mechanical properties depend on thestent geometry and structure. These examples will serve as a motivation to derive anoverall stress-strain relationship (pressure-displacement) of the “Law of Laplace”–typein terms of the geometric parameters of the stent, presented in section 7. We note herethat although the magnitude of the global radial or longitudinal displacement of theentire stent in these examples may be greater than 10%, the maximal displacementof the stent struts is considerably smaller. This is discussed in Appendix A.

In the first two examples we show that the magnitudes of the radial and longitudi-nal displacement of a stent under a uniform pressure load are larger if the stent is notmaximally expanded. Stents that are expanded to a larger radius are stiffer. Thus, inendovascular procedures in which stents are deployed to the regions exhibiting veryhigh pressure loads such as, for example, the annuli of the aortic heart valves in theaortic valve replacement, balloon expansion of the annular stent should be performedin a way that would ensure maximal stent expansion.

Example 6.1. A stainless steel stent (316L) with 8 vertices in the circumferentialdirection and 7 vertices in the longitudinal (axial) direction is considered. The lengthof each strut is 6mm. The stent has been expanded to the radius of 1cm into its refer-ence configuration. The stent is subject to a uniform pressure load of 0.5 atmosphereapplied to the exterior wall. As a result, the stent deforms and exhibits both the axialand longitudinal displacement. The maximum radial displacement is assumed at theendpoints of the stent, and it is equal to 1.52656mm (15% of the reference configura-tion). Figures 8, 9, and 10 show the magnitude of the radial and axial displacement,the total displacement and magnitude of rotation of cross section, and contact mo-ment and contact force, respectively. Negative displacement in Figure 8 correspondsto compression, while positive displacement corresponds to expansion. Notice thatthe maximum radial and longitudinal displacements occur at the endpoints of a stent,and that the largest cross-section rotation occurs at the middle of the strut with themaximum cross-section rotation occurring for the end struts. Additionally, Figure 10left shows that the largest contact moments occur at the stent’s vertices with themaximal contact moments occurring at the end vertices. Figure 10 right shows thatthe maximal contact force occurs at the end struts of a stent.

Example 6.2. All the stent parameters in this example as well as the magnitudeof the pressure load are the same as those in Example 6.1. The only difference is thereference configuration to which the stent has been expanded: The stent is now ex-panded to 0.8 of the reference configuration in the previous example, i.e., to 8mm. Ourresults below show that the maximal radial displacement is now 1.88614mm which is23.5% of the reference configuration. This is in contrast with the 15.2% correspond-ing to the maximal radial displacement in Example 6.1. Thus, we conjecture that thelarger the expansion of a stent, the larger the pressure loads that can be supportedby the stent. See Figure 11.

The next two examples show that nonuniform pressure loads cause higher stentdeformations. This corresponds to, for example, a situation when a stent is inserted in



0.00152656

0.00112135

0.010.00

0.010.000.010.020.03

0.0

0.00

0.01

0

0.00121413

0.010.00

0.010.000.010.020.03

0.0

0.00

0.01

Fig. 8. The magnitude of the radial (left) and axial (right) displacement for a stent in Exam-ple 6.1.

0

0.00195051

0.010.00

0.010.000.010.020.03

0.0

0.00

0.01

0

0.177471

0.010.00

0.010.000.010.020.03

0.0

0.00

0.01

Fig. 9. The magnitude of the total displacement (left) and rotation (right) of cross sections fora stent in Example 6.1.

0

0.000200332

0.010.00

0.010.000.010.020.03

0.0

0.00

0.01

0

0.0636342

0.010.00

0.010.000.010.020.03

0.0

0.00

0.01

Fig. 10. The magnitude of the contact moment (left) and the contact force (right) for a stentin Example 6.1.

0.00192311

0.001411060.010

0.0050.000

0.0050.010

0.000.010.020.030.01

0.00

0.000

0.005

0.010

0

0.001090220.010

0.0050.000

0.0050.010

0.000.010.020.030.01

0.00

0.000

0.005

0.010

Fig. 11. The magnitude of the radial (left) and axial (right) displacement for a stent in Exam-ple 6.2.

a vessel lumen with either high diameter gradients or nonaxially symmetric geometrywhich can occur due to, for example, plaque deposits that have not been uniformlypushed against the wall of a diseased artery during balloon angioplasty. We will showthat in this case the global stiffness of a stent is smaller than that of a stent underuniform pressure loads. Thus, a stent deployed to the lumen of an artery whose



0.010.00

0.010.000.010.020.03

0.01

0.00

0.01

0.00239282

0.00185101

0.010.00

0.010.000.010.020.03

0.01

0.00

0.01

0.00239282

0.001851010.00

0.01

0.02

0.03

0.01 0.00 0.01

0.01

0.00

0.01

0

0.0002157910.00

0.01

0.02

0.03

0.01 0.00 0.01

0.01

0.00

0.01

Fig. 12. The top figure on the left shows the four points at which the pressure load is applied tothe stent described in Example 6.3. The remaining three figures show the deformation of the stentsuperimposed over the reference configuration shown in grey. The stent struts are colored based onthe magnitude of the radial displacement.

0.010.00

0.010.000.010.020.03

0.01

0.00

0.01

0.00301197

0.000512745

0.010.00

0.010.000.010.020

0.01

0.00

0.01

.03

Fig. 13. The figure on the left shows the eight points at which the pressure load is appliedto the stent described in Example 6.4. The figure on the right shows the deformation of the stentsuperimposed over the reference configuration shown in grey. The stent struts are colored based onthe magnitude of the radial displacement.

diameter exhibits high gradients will deform more than a stent deployed in an arterywith nearly constant diameter and axially symmetric geometry.

Example 6.3. A stent from Example 6.1 is considered with the reference radiusof 1cm. A total load, which corresponds to the total forcing of uniform pressureof 0.05 atmosphere, is applied to the four points shown in Figure 12. Our numeri-cal simulations, presented in Figure 12, show that the maximal radial displacementequals 2.39282mm, which is almost two times the displacement achieved at the tentimes greater forcing applied uniformly. Additionally, the deformation, as expected,is nonuniform: The stent is compressed in the direction of the applied force andexpanded in the direction perpendicular to the applied force.

Example 6.4. A stent from Example 6.1 is considered with the reference radiusof 1cm. A force is applied to the eight points in the middle of the stent, shown inFigure 13. The force applied at each of the eight points is equal to 1/8 of the totalforce that corresponds to the uniform pressure load of 0.5 atm. The absolute value



0.00172265

0.00235094 0.0050.0000.005

0.000.010.020.03

0.005

0.000

0.005

Fig. 14. A stent from Example 6.1 is exposed to the uniform pressure of 0.5 atm applied tothe interior surface of the stent. The figure shows the dogboning effect (flaring of the proximal anddistal ends of a stent) typically observed during balloon expansion of a stent. See Example 6.5.

of the calculated maximal radial displacement is equal to 3.01197× 10−3m which isabout two times the maximal radial displacement from Example 6.1.

Example 6.5. The stent from Example 6.1 with the reference radius of R = 0.5cmis exposed to the uniform pressure of 0.5 atm applied to the interior surface of thestent. Figure 14 shows that our model captures the “dogboning” effect correspondingto the flaring of the proximal and distal ends of a stent, observed during balloonexpansion of the stent.

In the last sequence of six examples we study how the pressure-displacement rela-tionship for a given, uniform stent depends on the following geometric and mechanicproperties of a stent: the Young’s modulus E, the shear modulus μ, the thicknessof each stent strut t, the width of each stent strut w, the stent reference radius R,and the stent reference length L. In all of those examples the stent is subject to auniform pressure load of 0.5 atmospheres applied to the exterior surface of the stent.We begin with Example 6.6 which will serve as a benchmark against which the resultsfrom Examples 6.7 through 6.12 will be compared.

Example 6.6. In this example we begin with a stent with the parameter valuesshown in Table 1. The stent is subject to a uniform pressure load of 0.5 atmospheresapplied to the exterior surface of the stent. Figure 15 shows the deformation ofthe stent with the colors of the struts corresponding to the magnitude of the totaldisplacement. The calculated radial displacement at the middle vertex of the stentis equal to 3.35× 10−4m.

In the next two examples we vary the Young’s modulus E and the shear modulusμ independently to see how the displacement of a stent is influenced by those twoparameters independently. The choice of the two parameters was motivated by thefact that they appear in the coefficients of the curved rod model. Even though inan isotropic material, the Young’s modulus and the shear modulus are linked by μ =E/2 ∗ (1+ ν), the main purpose of the two numerical tests presented in Examples 6.7

Table 1

Parameter values for a stent in Example 6.6.

nO Number of vertices in circumferential direction 8nL + 1 Number of vertices in axial direction 7

E Young’s modulus of elasticity 2.1× 1011Pa

μ Shear modulus 8.3× 1010Pat Thickness of each strut 0.0001 mw Width of each strut 0.0001 mR Stent reference radius 0.01 mL Stent reference length 0.018 m



0

0.0007018850.01

0.00

0.01

0.0000.0050.0100.0150.020

0.01

0.00

0.01

Fig. 15. Stent from Example 6.6. The stent struts are colored based on the magnitude of totaldisplacement. The reference configuration is shown in grey.

0

0.00134469 0.01

0.00

0.01

0.0000.0050.0100.0150.020

0.01

0.00

0.01


and 6.8 was to show the poor influence of the torsion. This is later confirmed in thederivation of the effective pressure-displacement relationship in section 7.

Example 6.7. Consider a stent with the parameter values from Table 1, exceptfor the Young’s modulus which is now taken to be one-half of the Young’s modulusfrom Table 1, i.e., E = 1.05×1011Pa. Under the applied uniform pressure of 0.5 atmo-spheres, our simulation provides the radial displacement at the middle vertex of thestent equal to 6.65× 10−4m which is around twice the size of the radial displacementfrom the previous case. The corresponding deformed stent is shown in Figure 16.

Example 6.8. Consider a stent with the parameter values from Table 1 except forthe shear modulus which is now taken to be half of the shear modulus from Table 1,i.e., μ = 4.15 × 1010Pa. The stent is subject to the uniform pressure load of 0.5atmospheres. Our simulation provides the radial displacement at the middle vertex ofthe stent equal to 3.38× 10−4m, which is almost the same as the radial displacementin Example 6.6. The corresponding deformed stent is shown in Figure 17. Thus,changing the shear modulus μ does not seem to influence the stent displacement tothe leading order.

Example 6.9. Consider a stent with the parameter values from Table 1 exceptfor the thickness of each strut which is now taken to be half of the thickness fromTable 1, i.e., t = 0.00005m. The stent is subject to the uniform pressure load of 0.5atmospheres. Our simulation provides the radial displacement at the middle vertexof the stent equal to 6.96 × 10−4m, which is double the radial displacement fromExample 6.6. The corresponding deformed stent is shown in Figure 18.

Example 6.10. Consider a stent with parameter values from Table 1 except forthe width of each strut which is now taken to be half of the width from Table 1, i.e.,w = 0.00005m. The stent is subject to the uniform pressure load of 0.5 atmospheres.



0

0.00069939 0.01

0.00

0.01

0.0000.0050.0100.0150.020

0.01

0.00

0.01


0

0.00144841 0.01

0.00

0.01

0.0000.0050.0100.0150.020

0.01

0.00

0.01


0

0.00271467 0.01

0.00

0.01

0.0000.0050.0100.0150.020

0.01

0.00

0.01


Our simulation provides the radial displacement at the middle vertex of the stentequal to 1.28× 10−3m, which is four times the radial displacement from Example 6.6.The corresponding deformed stent is shown in Figure 19.

Example 6.11. Consider a stent with parameter values from Table 1 except forthe reference total length of the stent which is now taken to be half of the referencelength from Table 1, i.e., L = 0.09m. This means, in particular, that the total lengthof the struts is now smaller than the total strut length for a stent with parametersdescribed in Table 1. All the other parameters are the same as those presented inTable 1. The stent is subject to the uniform pressure load of 0.5 atmospheres. Oursimulation provides the radial displacement at the middle vertex of the stent equal to6.34×10−5m which is smaller than the radial displacement of the stent with referencelength from Table 1. Thus, shorter stents with the same number of vertices are stifferthan longer stents. The corresponding deformed stent is shown in Figure 20.



0

0.000208205 0.01

0.00

0.01

0.0000.0050.010

0.01

0.00

0.01

Fig. 20. Stent from Example 6.11. The stent struts are colored based on the magnitude of thetotal displacement. The reference configuration is shown in grey.

0

0.000270268 0.005

0.000

0.005

0.0000.0050.0100.0150.020

0.005

0.000

0.005


Example 6.12. Consider a stent with parameter values from Table 1 except forthe reference radius of the stent which is now taken to be half of the reference radiusfrom Table 1, i.e., R = 0.005m. Again, this means, in particular, that the total lengthof stent struts is smaller than the total length of the stent struts from Example 6.6.All the other parameters are the same as those in Table 1. The stent is subject tothe uniform pressure load of 0.5 atmospheres. Our simulation provides the radialdisplacement at the middle vertex of the stent equal to 1.74× 10−4m which is abouthalf the radial displacement of the stent from Example 6.6. Thus, the smaller thereference radius of a stent, the stiffer the stent. The corresponding deformed stent isshown in Figure 21.

7. An effective model for the global behavior of a uniform stent. Thegoal of this section is to derive a simple model, in the form of the Laplace law, forthe global behavior of the entire stent. In particular, we are interested in deriving arelationship between the applied uniform pressure and the following displacements:(1) the radial displacement at the middle of a stent, (2) the longitudinal displacementat the endpoints of a stent.

Recall that Laplace law for a linearly elastic membrane reads [7]

p =Et

(1 − σ2)R2u.

For small deformations and small deformation gradients, we aim at describing theglobal behavior of a stent by looking for a relationship of the same form

p = Ku,



where p is the applied uniform pressure, u is the stent displacement (radial or longi-tudinal), and K is the proportionality constant that depends on the following param-eters:

• material properties described by E, μ;• geometry of the strut cross section described by w, t; and• overall geometry of the stent described by R, L, nO, and nL.

Thus, the aim is to identify the constantK in terms of the mechanic and geometric pa-rameters of the stent. We will first obtain an expression for K by combining the finiteelement method–based numerical experiments and the least square approximation ofthe experimental data and then derive the same formula using the minimization (ofenergy) formulation of the stent problem and a simple geometric argument.

We begin by assuming the following homogeneous dependence of K on the pa-rameters in the problem:

p = kEα1μα2wα3tα4Rα5Lα6u,(7.1)

where we need to identify the powers αi and the constant k, where k = k(nO, nL) willembody the dependence of the pressure-displacement relationship on the geometricalparameters nO and nL. As we shall see below, our numerical investigation leadingto the values of the powers αi produced amazing results implying integer values ofthe powers αi that provide excellent agreement with the (numerical) experiments andwith the effective equations derived from the energy equality.

To determine the powers αi we first rewrite the problem in linear form by takingthe logarithm of (7.1) to obtain

log p = log k + α1 logE + α2 logμ+ α3 logw + α4 log t+α5 logR+ α6 logL+ log u.

(7.2)

We then use the linear least square method to approximate the data obtained usingthe finite element method, described in the previous section. We ran the finite elementmethod on the following set of parameters:

E ∈ 2.1× 1011 × {0.8, 0.85, 0.9, 0.95, 1, 1.05, 1.1, 1.15, 1.2}Pa,μ ∈ 8.31× 1010 × {0.85, 0.9, 0.95, 1, 1.05, 1.1, 1.15}Pa,w ∈ 0.0001× {0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.1, 1.2, 1.3, 1.4, 1.5}m,t ∈ 0.0001× {0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.1, 1.2, 1.3, 1.4, 1.5}m,R ∈ 0.01× {0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, 1, 1.05, 1.1, 1.15, 1.2}m,L ∈ 0.018× {0.8, 0.9, 1, 1.1, 1.2, 1.3, 1.4, 1.5}m.

For each set of values of {E, μ,w, t, R, L} and a given, fixed, pressure p, the corre-sponding displacement was recovered by the finite element method. The linear leastsquare method is then used to approximate the obtained numerical results using afunction of the form (7.2). For the pressure-longitudinal displacement relationshipthe results of the least square method are shown in Table 2. For the pressure-radialdisplacement relationship the results of the least square method are shown in Table 3.

Conclusion 1. The least square method suggests the following values of αi:

α1 = 1, α2 = 0, α3 = 2, α4 = 1.

Notice that α2 = 0 confirms the poor influence of the torsion, as suggested by Exam-ples 6.7 and 6.8 in section 5.



Table 2

Parameters for the pressure-longitudinal displacement relationship.

log kl α1 α2 α3 α4 α5 α6

nO = 6, nL = 6 −4.114 0.990 0.009 1.973 1.030 −2.260 −1.694nO = 8, nL = 6 −3.868 0.995 0.004 1.983 1.019 −1.995 −1.955nO = 12, nL = 6 −3.600 0.998 0.001 1.992 1.009 −1.631 −2.324

Table 3

Parameters for the pressure-radial displacement relationship.

log kr α1 α2 α3 α4 α5 α6

nO = 6, nL = 6 −3.439 0.987 0.012 1.940 1.053 −1.275 −2.678nO = 8, nL = 6 −3.419 0.991 0.008 1.950 1.047 −1.005 −2.945nO = 12, nL = 6 −3.484 0.997 0.002 1.978 1.024 −0.634 −3.321

Table 4

Value of parameters in formula (7.3) with u corresponding to the longitudinal displacement

log kl α1 α2 α3 α4 α5 α6

nO = 6, nL = 6 −1.252 0.990 0.009 3.973 1.030 −1.012 −0.988nO = 8, nL = 6 −0.997 0.995 0.004 3.983 1.019 −1.006 −0.993nO = 12, nL = 6 −0.643 0.998 0.001 3.992 1.009 −1.002 −0.997

To fix the last two powers, we rescale the pressure by the square of the approximatecross-sectional area. This is the total area of the stent material, assuming a linearapproximation of the curved stent struts (using le instead of ls). The total area isgiven by

A = 2nLnOwle.

This proved to be the correct approach, since, as we shall see below, this producedthe values of new parameters α5 and α6 independent of nO. Thus, we reformulate theproblem by looking for the pressure-displacement relationship of the form

p =k

A2Eα1μα2wα3tα4Rα5Lα6u, A = 2nLnOwle.(7.3)

The least square method was then run for the approximation of the form (7.3), andthe relative �2 error between the FEM simulation and formula (7.3) was calculatedbased on the following formula:

Error =1

N

∑Data Set

(FEM simulation − result using (7.3)

result using (7.3)

)2

,

where N is the number of points in Data Set. Table 4 shows the results obtained usingthe least square method.

Based on these results we conclude the following.Conclusion 2. The assumption (7.3) on the pressure-longitudinal displacement

relationship leads to the values of αi, i = 1, . . . , 6 that approximate the followinginteger values:

α1 = 1, α2 = 0, α3 = 4, α4 = 1, α5 = −1, α6 = −1.(7.4)

The relative error is then obtained by comparing the data obtained using the FEMmethod and the data obtained using the values from Table 4 in formula (7.3). Therelative errors are presented in Table 5.



Table 5

Relative errors for the least square method results presented in Table 4.

relative �2 errorN

nO = 6, nL = 6 4.5711 × 10−6

nO = 8, nL = 6 3.649× 10−6

nO = 12, nL = 6 2.26763 × 10−6

Table 6

Value for the parameters in formula (7.3) with u corresponding to the radial displacement.

log kr α1 α2 α3 α4 α5 α6

nO = 6, nL = 6 −0.578 0.987 0.012 3.940 1.053 −0.026 −1.972nO = 8, nL = 6 −0.548 0.991 0.008 3.950 1.047 −0.016 −1.983nO = 12, nL = 6 −0.527 0.997 0.002 3.978 1.024 −0.005 −1.994

Table 7

Relative error for the least square method results presented in Table 6.

relative �2 errorN

nO = 6, nL = 6 8.51638 × 10−6

nO = 8, nL = 6 2.94811 × 10−6

nO = 12, nL = 6 2.52609 × 10−6

Similarly, for the pressure-radial displacement relationship the results of the leastsquare method are presented in Table 6.

Based on those results we conclude the following.Conclusion 3. The assumption (7.3) on the pressure-radial displacement rela-

tionship leads to the values of αi, i = 1, . . . , 6 that approximate the following integervalues:

α1 = 1, α2 = 0, α3 = 4, α4 = 1, α5 = 0, α6 = −2.(7.5)

The �2 relative error for the results presented in Table 6 is shown in Table 7.Based on the above results we obtain the following.Conclusion 4. In the pressure-displacement relationship p = Ku, where u is

either the radial displacement ur in the middle of a stent or the longitudinal dis-placement ul of the endpoints of a stent, the coefficient K depends on the Young’smodulus of each strut E, on the width w and thickness t of each strut with rectangularcross section, and on the overall length L and reference radius R of the stent, via thefollowing relationship:

• for the longitudinal displacement at the endpoints of a stent

p =klA2

Ew4t

RLul,(7.6)

• for the radial displacement at the middle point of a stent

p =krA2

Ew4t

L2ur.(7.7)

Here A is the total area of the struts, and kl and kr are the coefficients that dependon the number of vertices in the circumferential direction nO and on the number of



Table 8

Parameters C, β1, and β2 for the pressure-longitudinal displacement relationship.

logC β1 β2relative �2 error

N

K −4.434 2.046 2.022 0.000115332

Table 9

Parameters C, β1, and β2 for the pressure-radial displacement relationship.

logC β1 β2relative �2 error

N

K −3.901 0.032 3.947 0.000161531

vertices in the longitudinal direction nL + 1. The precise dependence of kl and kr onthese parameters will be analyzed below.

Notice that formulas (7.6) and (7.7) differ by a factor of R/L. This is in agreementwith the asymptotic expansions (with respect to R/L) leading to the rod equationswhich imply that the longitudinal displacement of a rod is by one order of magnitudesmaller than the transverse displacement [9], namely,

ul ≈ R

Lur.(7.8)

Notice also that coefficient K does not depend on the shear modulus μ. This isto be expected, as noticed in Example 6.8, since the influence of the torsion of eachcross section onto the overall pressure-displacement relationship of the stent shouldbe of a lower order of magnitude, not captured in the leading-order behavior, usingapproximations (7.6) and (7.7).

In the last step of this procedure we determine how the coefficients kl and kr informulas (7.6) and (7.7), respectively, depend on the geometrical parameters nO andnL. This will fully determine the dependence of K on the parameters in the problem.We make the following assumption:

p =C

A2nβ1

O nβ2

L Eα1μα2wα3tα4Rα5Lα6u,

where α1, . . . , α6 are given by either (7.4) or (7.5), depending on the considered case.Thus, we have assumed, in a similar fashion as before, that kr and kl depend on nO

and nL through the powers β1 and β2 of nO and nL, respectively. The coefficient Cand the powers βi will depend on whether the longitudinal or the radial displacementis being considered. The results shown in Tables 8 and 9 show the values of thecoefficients C, β1, and β2 with the corresponding relative errors, calculated for thefollowing sets of parameter values:

E ∈ 2.1× 1011 × {0.8, 1, 1.2}, μ ∈ 8.31× 1010 × {0.85, 1, 1.15},w ∈ 0.0001× {0.5, 1, 1.5}, t ∈ 0.0001× {0.5, 1, 1.5},R ∈ 0.01× {0.8, 1, 1.2}, L ∈ 0.018× {0.8, 1, 1.2}.

Conclusion 5. The least square method indicates that the values of the param-eters β1 and β2 are given by β1 = 0, β2 = 4 for the pressure-radial displacementrelationship, and by β1 = 2, β2 = 2 for the pressure-longitudinal displacement rela-tionship.



Table 10

Proportionality constant Cl for the longitudinal displacement of a stent at end points.

logClrelative �2 error

N

K −4.370 0.000172441

Table 11

Proportionality constant Cr for the radial displacement of a stent at the middle point.

logCrrelative �2 error

N

K −3.916 0.00200469

Thus we have obtained the following.Main result. Based on the data obtained using the finite element method simula-

tions of the stent deformation under the uniform pressure applied to the interior or theexterior surface of a stent, the following simplified pressure-displacement relationshipis obtained using the least square method approximation of the data for the longitu-dinal displacement at the endpoints of a stent ul, and for the radial displacement ofthe middle point of a stent ur:

p = ClEw4tn2

On2L

RLA2ul =

Cl

4

Ew2t

RLl2eul,(7.9)

p = CrEw4tn4

L

L2A2ur =

Cr

4

Ew2tn2L

L2l2en2O

ur,(7.10)

where

l2e = 4R2 sin2π

2n0+

(L

nL

)2

.

Here E is the Young’s modulus of the stent struts, w and t are the width and thethickness of the stent struts, R and L are the reference radius and length of theentire stent, nO and nL + 1 are the numbers of vertices in the circumferential andlongitudinal directions, respectively, and

A = 2nOnLwle(7.11)

is the area of the stent struts, where le is the distance of the endpoints of a strut(approximate strut length). Estimates for the values of the constants Cl and Cr,using the least square method, assuming the values of β1 and β2 as in Conclusion 5,are shown in Tables 10 and 11.

Derivation of the effective equations using the energy formulation. We can recoverformulas (7.9) and (7.10) by studying the leading-order behavior of the energy ofproblem (4.8), obtained from its weak formulation assuming linear approximation ofthe curved stent struts.

Namely, the total energy of problem (4.8) is given by the following integral:

IE :=1

2

∑i,j,k

∫ ls

0

EIn((uki,j)

′′n

)2+ EIb

((uki,j)

′′b

)2+ μK

((τki,j)

′)2 ds

−∑i,j,k

∫ ls

0

(fki,j)t(u

ki,j)t + (fk

i,j)n(uki,j)n + (fk

i,j)b(uki,j)bds.(7.12)



n

b

P1 P2

P3

2R

sin

4

ls

b

Fig. 22. Left: One circumferential ring of stent struts showing the normal (radial) and a bi-normal vector on one of the struts. Right: Right angle triangle P1, P2, P3 with sides of lengthls (strut), Δ (parallel with the axis of symmetry), and 2R sin(φ/4) where φ is the angle defined insection 2, Figure 5.

The solution of problem (4.8)–(4.11) is the minimum of functional IE given in (7.12)over the subspace determined by the conditions (4.9)–(4.11). In the energy integral(7.12) moments of inertia are given by

In =1

12wt3 and Ib =

1

12w3t.(7.13)

To study the leading-order behavior of (7.12), we introduce the nondimensionalvariables. To simplify notation, we first drop the i, j, k subscripts and superscriptsand consider a generic straight rod whose displacement and torsion we denote by(ut, un, ub, τ). Introduce the nondimensional variables ut, un, ub, and τ so that

ut(lsz) = Utut(z),

un(lsz) = Unun(z),

ub(lsz) = Ubub(z),

τ(lsz) = τ(z),(7.14)

which are defined on the unit interval z ∈ (0, 1) where s = lsz.Consider one circumferential ring of struts as shown in Figure 22 left. Denote

by U locl the local (longitudinal) displacement of one circumferential ring of struts in

the axial (longitudinal) direction. Assuming that each circumferential ring suffersthe same local displacement U loc

l , the total longitudinal displacement Ul of a stentconsisting of nL rings is given by Ul = nLU

locL . From the right angle triangle P1,

P2, P3 shown in Figure 22 right, the bi-normal displacement Ub of a stent strut, thenormal displacement Un of a stent strut, and the local longitudinal displacement U loc

l

are related by the following:

Ub =ls

2R sin(φ/4)U locl ,(7.15)



U locl =

R

L4 sin2(φ/4)n2

LUn,(7.16)

where φ is the angle defined in section 2, Figure 5. Equation (7.16) is obtained fromthe right angle triangle P1, P2, P3, shown in Figure 22 from which

Δ2 + 4R2 sin2(φ/4) = l2s.

Deformation of this triangle due to the negative local longitudinal displacement U locl

and positive normal (radial) displacement Un (caused by the interior pressure loading)gives rise to

(Δ− U locl )2 + 4(R+ Un)

2 sin2(φ/4) = l2s .

Here it was assumed that the right angle triangle is deformed (approximately) intoanother right angle triangle with sides Δ−U loc

l , 2(R+Un) sin(φ/4), and ls. AssumingUn small, the first term in the Taylor series expansion of U loc

l in terms of Un is givenby

U locl =

4R sin2(φ/4)√l2s − 4R2 sin2(φ/4)

Un + · · · .

Since the expression in the denominator equals Δ, we get

U locl =

4R sin2(φ/4)

ΔUn + · · · .

Finally, since Δ = L/nL we recover (7.16). Thus, from the geometric considerationswe have shown that the radial displacement corresponding to Un and the longitudinaldisplacement Ul are related to the leading order via

Ul =R

L4 sin2(φ/4)n2

LUn.(7.17)

This can be simplified even further by noticing that the number of circumferentialpoints in a stent nO is typically greater than three, nO ≥ 3. Recalling that φ = 2π/nO

we see that φ/4 = π/(2nO) ≤ π/6. Thus, the leading-order behavior of sin2(φ/4) isgiven by

sin2(φ/4) =π2

4n20

+ · · · .(7.18)

Using this approximation we get that the leading-order relationship between Ul andUn is given by

Ul =π2

n2O

R

LUn.(7.19)

This corresponds to the radial-longitudinal displacement relationship in terms of R/L,presented in formula (7.8).

We now turn back to the energy integral (7.12) to show that formulas (7.9) and(7.10) can be derived from (7.12) by taking into account only the term with the second-order derivative with respect to s of the bi-normal component of the displacement.



Fig. 23. Comparison between the reference strut (in black) and deformed strut (in red) viewedfrom the side (left picture) and from the top (right picture) indicating the normal displacement (leftpicture) and the bi-normal displacement (right picture).

This means that, to the leading order, the second-order derivative with respect to sof the normal displacement and the first-order derivative of the torsion of the strut’scross section are negligible with respect to the second-order derivative of the bi-normaldisplacement, under the uniform radial loading of a stent. This is reasonable. Namely,this means that the change in the curvature of a stent strut in the bi-normal direction islarger than the other contributions. Indeed, the most “visible” stent strut deformationis near the endpoints of a strut, at the vertices where the struts meet. This is, forexample, visible in Figure 23 right where the curvature of the deformed strut, shownin red, is visibly different from the curvature of the undeformed strut, shown in black.The change in the strut curvature in the normal direction, shown in Figure 23 left isvisibly smaller. Figure 23 left shows a comparison between a deformed strut, shown inred, and the reference strut, shown in black, viewed from the side (bi-normal direction)indicating the deformation of the strut in the normal direction.

Thus, we begin by calculating the term which contains u′′b on the left-hand sideof (7.12):

EIb(u′′b )

2 = Etw3

(Ub

1

l2su′′b

)2

.(7.20)

Now, by using (7.15) and (7.16) we obtain

Ub = 2ls sin(φ/4)nL1

LUn.

By plugging this equation into (7.20) we get

EIb(u′′b )

2 =Etw3

L2l2sn2L sin2(φ/4)U2

n(u′′b )

2.(7.21)

We now take into account the right-hand side which incorporates the forcing in thenormal direction which is given by the pressure p applied to the stent strut of widthw, namely, pwUn. By neglecting the lower order terms we conclude that coefficientsof the leading-order terms should be proportional, with a nondimensional constant ofproportionality C, namely,

p = CEtw2

L2l2sn2L sin2(φ/4)Un.(7.22)



Now take into account that A = 2n0nLwls, as stated in (7.11), to obtain

p = CEtw4

L2A2n4L4n

20 sin

2(φ/4)Un.(7.23)

By taking into account the leading-order approximation of sin2(φ/4) given by (7.18),we get from (7.23) the following pressure-radial displacement relationship:

p = CEtw4n4

L

L2A2Un,(7.24)

which is exactly formula (7.10).By substituting (7.19) into (7.24) one obtains

p = CEtw4n2

Ln2O

LRA2Ul,(7.25)

which is exactly formula (7.9).Remark 7.1. The fact that the leading-order energy behavior appears to be

determined by the stent strut deformation in the bi-normal direction indicates thatusing the curved rod model, which captures stent strut deformation in all spatialdirections, in contrast with a beam model, which captures deformation only in thedirection of the forcing, provides a superior one-dimensional strut approximation overthe conventional beam theory in the stent application at hand.

Consequences of formulas (7.10) and (7.9) can be summarized as follows.Corollary 7.1. In the production of a stent with uniform geometry, the radial

stiffness of a stent increases with the higher Young’s modulus and thickness of the stentstruts, and with the width of the stent struts. The radial stiffness of a stent decreaseswith the larger reference length L. The precise dependence on all the parameters isgiven by formula (7.10).

Corollary 7.2. In the production of a stent with uniform geometry, the longi-tudinal stiffness of a stent increases with the higher Young’s modulus and thicknessof the stent struts and with the width of the stent struts. The longitudinal stiffness ofa stent decreases with the increase in the total stent length L and with the increaseof the reference radius R. The precise dependence on all the parameters is given byformula (7.9).

Appendix A. In Example 6.1 we studied the behavior of a stent with referenceradius 1 cm exposed to a uniform pressure load of 0.5 atm applied to the exteriorsurface of a stent, causing compression. The maximum radial displacement of theentire stent was assumed at the endpoints, and it was equal to 15% of the referenceconfiguration. In this appendix we show that an arbitrary strut in this stent underthis loading deforms much less than 15% and well within the realms of linear theoryindicating that for the application discussed in this paper, using the linear theory ofelasticity to describe the deformation of stent struts is appropriate. In order to do thiswe chose a strut which experiences maximal deformation in the sense of the contactmoments (experiencing maximal loading). This is the strut with vertices i = 5 andj = 13 corresponding to one of the end struts. We calculated the total deformationof this strut. Figure 23 right shows a comparison between the reference configurationof the strut (shown in black) and the deformed strut (shown in red). The two are su-perimposed so that the deformed strut was translated to the position of the deformedstrut in the least square sense. Figure 23 left shows the reference and the deformed



strut from the side (i.e., the bi-normal direction of the reference strut). The maximaldisplacement of the strut divided by the strut length was calculated to be equal to 6.5%which is well within the realms of linear theory. This indicates the appropriateness ofthe use of linear theory to study compression and expansion of an already expandedstent under the physiologically relevant cyclic loading of approximately 0.5 atm.

Appendix B. To compare the performance between the 1D curved rod modelemployed in this manuscript to describe the mechanical properties of stent struts,with a 3D finite element method approximation of a curved rod using 3D linearelasticity, we consider here a semicircular rod of radius 8 × 10−4m (middle curveradius), and with a square cross section of thickness 2.5× 10−5m; see Figure 24. Thesemicircular rod with these dimensions has approximately the same aspect ratio asthe stent struts considered in Example 6.1 of this manuscript. Namely, the aspectratio of the semicircular rod is

ε =w

ls=

2.5× 10−5

π × 8× 10−4= 0.1× 10−1 = 10−2,

while the aspect ratio for the struts of the stent considered in Example 6.1 is ε(Ex1) =1.6 × 10−2m. (In [9] it was shown mathematically that the 1D curved rod modelapproximates the 3D problem in the sense that the 3D displacement converges to thesolution of the 1D problem in the H1 sense as the aspect ratio ε→ 0.)

The semicircular rod considered in this appendix was exposed to the loading of aforce with two components different from zero:

f = (fx, fy, fz) = (0, 109,−1011)N/m3,

where the y-direction, as shown in Figure 24, points toward the viewer, and thenegative z-direction points downward. The rod was fixed at the endpoints with asquare basis assuming the homogeneous Dirichlet boundary conditions.

A three-dimensional FEM approximation of the rod was performed using freefem3d(http://www.freefem.org/ff3d/) with 344259 tetrahedral elements. This gave rise toa system matrix of dimension 217017. The same problem was solved using a FEMapproximation of the 1D curved rod model, presented in this manuscript, with 100discretization points along the middle curve of the rod giving rise to a system matrixor dimension 606. Figures 24 and 25 show a comparison between the 3D (left) and 1D(right) approximations of the deformations of the rod. The pictures are colored basedon the magnitude of the y component of the displacement. The difference between thetwo approximations is 0.5% of the displacement with the maximum y displacement

0.

0.0000150821

0.000020.000000.000010.00002

0.0005 0.0000 0.0005

.0000

.0002

.0004

.0006

.0008

Fig. 24. The 3D (left) and 1D (right) simulations of a curved rod deformation. The referenceconfiguration is shown in grey. The rod is colored based on the magnitude of the y component ofthe displacement. The difference between the two calculations is less than 8 × ε−4 where ε = 10−2

is the aspect ratio of the rod.



0.

0.0000150821

Fig. 25. Side view of the semicircular rod shown in Figure 24.

equal to 1.5 × 10−5m. This is of order 8 × ε4 where ε = 10−2 is the aspect ratio ofthe rod. The difference in the other two components of the displacement were lessthan or equal to the reported error of the y-displacement, thereby indicating excellentapproximation by the 1D curved rod model at a much smaller computational cost.

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Documents

SIAM J. A MATH c Vol. 70, No. 6, pp. 1922–1952djelatnici.unizd.hr/~makosor/SIAM_stent.pdf · their design and performance. Even though a lot of attention has been devoted in cardiovascular